Burali–Forti paradox

The Burali–Forti paradox refers to the theorem that there is no set containing all von Neumann ordinals. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes).